Perfectly Matched Layer Methods For Solving Electromagnetic Equations And Multi-symplectic Methods For Hamiltonian PDEs Systems

The inchoate development of electromagnetic theory and computational electromagnetics was firmly combined to telecommunication and radar technology, while restricted to military purpose at early period. Now people find its applications throughout earth science, life science, medical science, material science and information science and so on. The main problem in computational electromagnetics is to solve the electromagnetic equations (Maxwell equations), of which an important task is how to truncate the infinite domain into a finite domain.Since Berenger firstly proposed the Perfectly Matched Layer(PML) by the idea of perturbation in 1994, people have constructed a variety type of PML in different point of views. The key of this method is to introduce a specifically designed region (PML region) surrounding the computational region so that the electromagnetic field satisfies different Maxwell equations in different region and the outgoing wave can be effectively absorbed (or the reflective wave sufficiently small). As a result, the Engineers and mathematicians have applied various numerical methods to PML and observed its superiority over traditional methods.In this paper, we study the applications of PML method on Maxwell equations, which is extendable to the linear hyperbolic systems of first order. The main work of this thesis is:(1) Using the anisotropic medium formulation, which is proposed by Sacks, we identify a unified formulation for most existing models of PML method for two dimensional MaxwellAll the existing perfectly matched layer mehtods, such as Berenger's PML by perturbation. Chew and Weedon's stretching complex coordinate formulation, Zhao and Cangellaris's formula with source, Ziolkowski's equations by Lorentz model and so on can be reduced to the unified formulation. It is a powerful tool, for example, when discussing the well-posedness of the PML model, we only consider the unified formulation and specify the concrete case. (2) Consider the one-dimensional Maxwell equations:&E _ dH_ dH _ d. dt dx' dt d.and its corresponding finite Berenger PML equations:X 0,x < d,(2)aEa'd{d. t), t) = 0, t > 0the initial data E(x, 0) = Eq{x) and H(x, 0) = Hq(x) are assumed to be supported on x < 0 and the absorption profile a(x) is supported on x > 0, where .| ^ l,a0 / 0.Theorem 1 Assume that a(x) € C:(R). 0. then the system (2) has a unique solution (Ea'd,H"-d), which satisfiesTheorem 1 provides two conclusions: the numerical solution exponentially goes to the analytical solution in the computational domain x < 0 as the width of the PML or the absorption profile a(x) goes to infinite;next, the estimate (4) tells us that the reflection is small as the width of the PML sufficiently large, and the numerical solution decay exponentially in the PML region, therefore the method gives a good simulation of the electromagnetic field in the computational domain.(3) We construct the following up-winding scheme according to the continuous case:rr(T,d,nrc,d,n rjc.d,n TTa,d,nUi= 0,f d i AX d 30, (5)1 t rO.a.V,. t j-CT.u.rj t /-CT.a.71 "^ — 7— V . v ■ i — V ■At J J+1 Axthe boundary condition is V° 0. \Hj'd'n - H{Xj,tn)\ = O(Ax + At) + Coe-J?^)d^Moreover. Ej' '" and HJ' 'n decay exponentially in space, i.e.,\E°'d'n\ < Coe- & "?)?, \H°'d'n\ < Coe~& CT?'d?, 0 < j < p, n > 0, ^ ! + \H0(x)\}.o p{p + a x 0. /?, = -^--{4a* + 3(o-'jf-2cTjo-'j)! then for any 0 < k < Ar - 1, t > 0, we haveY^ h(Ej(t))2 < Coe-2^ f^ Yw/?,ere Co = IJBollo,?,. + II^o||o,/i>j w^ II ' llo.ft ?"^ II ? Ilo,;?* denote the discrete P-norm on the grids and its dual partitions respectively.Introduce the error functions ef{t) = E(x^t) - E°td(t) and ef+i(t) = H{xJ+ut) -)> and definethen we can show Theorem 4O(h2). then we haveTheorem 4 Under the assumptions of Theorem. 3. if ||eE(O)||o,/, = ||eff(O)||o,h-Some numerical experiments are presented, which validate the effectiveness of the finite difference schemes proposed in this dissertation.In the second part of this thesis, we study the multi-symplectic method for the Hamiltonian partial differential equations. After the pioneering work of Feng Kang and Ruth, an enormous intensive studies were launched on the symplectic integration of Hamiltonian systems in the 80th of last century. An important change of view-point came about from only considering the numerical simulation of a single trajectory shift to considering the numerical simulation of phase flow of a differential dynamical system through given numerical method. It is expected that the geometry of phase space in the problem should be preserved by a properly chosen numerical method. This view leads to the well-known geometric numerical integration and which is being under extensively study nowadays. It turned out that the geometric preservation properties of numerical methods not only produces better simulation of qualitative behavior, but also allows to obtain more accurate results for long-time integration than those obtained by the usual methods.More recently, Marsden and Bridges proposed the notion of Hamiltonian PDEs, i.e.. the multi-symplectic PDEs. It is found that some of the well-known nonlinear evolution equations, such as, Schrodinger equation, Sine-Gordon equation, Boussinisq equation. Korteweg de Vries equation and so on, all of them have the multi-symplectic structure. And then Marsden .Bridges and Reich have extended the symplectic integration method to the PDE case based on such multi-symplectic reformulations, i.e., which is known to be multi-symplectic integration.Our main work of this part is:(1) Multi-symplectic PDEs are denned byKdtz + Ldxz = VzS(z), (8)where z € R, 5 : Rn —-> R is some smooth function, and K, L are two nxn skew-symmetric constant matrices. There's plenty of theoretical results concerning such multi-symplectic reformulation, and among them the most remarkable result might be the existence of several local invariants, which are always hallmarks for conservative systems. For system (8), those local invariants are, namely, multi-symplecticity, the existence of energy and momentum conservation laws. Two kinds of definition of multi-symplectic integrator can be found in literatures, and in the present paper, we follow the point of view in [4].(where dt' andd^'1 are discretizations of the derivatives dt anddx) is called multi-symplectic. if it satisfies a discrete version of the multi-symplectic conservation law dt' u^+c^''Kk,i = 0. where uJkj = dzk,i A Kdzi^i. Kk,i = dz^.i A Ldzk.i- and zk,i satisfies the discrete variational equation Kd^'dzk,i + Ld^ldzkj = S"{zk,i)dzk,i.For the simplicity, we give a detailed discuss on the Multi-symplectic Hamiltonian PDEs by using the Hamiltonian nonlinear wave equation as an example. Five kinds of multi-symplectic reformulations are listed (see page 84), and the advantages and disadvantages are analyzed. By using the difference operators:5Izk,i ■=--------r--------> k,l+c ?es + hAi2)[-dtVk+c,l+c - FWk+cl+c)},Uk+c,l+c =Uk+c.l ? ej + TdtUk+c,l+c{A(1)}T.Pk+cj+c. =Pk^d ? el + TdtPk+c,i+c\A(l)]7\Vfc+c,J+c =vk+c,i ?ej + TdtVk+cu+c{A(2)}T,Wk+cj+c =wk+c.i ? el + TdtWk+c,i+c[A{2)}T,c =Pk,i+c + hbT[-dtUk+cj+c + Vfc+c.i+e],c =ul,l+i + hbT[-dtPk+cj+-c -+c],(15) c =wkj+c + hbT[-dtVk+c,i+c ~ '{%c =^lj+c - hbTdt\VkPk+c,i+i =Pk+d + tk+c,i+d,(16)t't+cHl =Vk+c.l + TdtVk+d+cb,+l =Wk+c,l + TdtWk+cl+cb-The scheme (13)-(16) can also be taken as a two-level finite difference method, with the discrete values ujT+1 [+i, Uk+c,i+i etc. (k = 0,1,..., X/h;I = 0,1,2,...) to be determined.Proposition 2 IflZ^-H1-2^ andTZ1-1^-TZ^ both are symplectic and satisfy the assumption (12), then the scheme (15)-(16) is multi-symplectic, and exhibits the discrete multi-symplectic conservation law such as the one given in (11).Using the above notations, the MSRK method be readily regarded as a special two-level finite difference scheme, with existing stability analysis of finite difference scheme by Fourier'method ( see page 92-93), and notation 8 = —, we obtain the following results:h Theorem 6 (1) When 0 < 9 < 1, Euler Box I. Ill and IV are stable;(2) When0 < 9 < 1, Euler Box II is stable;(3) For any 6 > 0, Euler Box V is unstable.Theorem 7 (1) When 9 > 0, Preissman Box I. Ill and IV are stable;(2) When 6 > 0, Preissmann Box II is stable.(2) Consider the linear multi-symplectic Hamiltonian PDEs, namely take 5(2) = I < z, Sz >, with S a symmetric matrix. Dispersive waves usually are recognized by z(x, t) = a.el^KX~ut\ where the constants k,lj and a are respectively, the wave number, frequency and amplitude. The amplitude vector a is determined by the initial data, whereas, the wave number k and the frequency uj have to be related by the dispersion relation.S) = 0, (17)where u>(k) is a real value function and u/'(k) ^ 0.We seek the numerical dispersive waves solution taken the following form(18)where A' is the numerical wave number and fi is the numerical frequency (sometimes called the Nyquist frequency) such that — tt < hK < tx, —ir < tQ < n.Theorem 8 The discrete multi-symplectic method for the linear Klein- Gordon equation possesses the following numerical dispersion relationdet(Jr(A",fl)K + S(A',n)L-S) = O, (19)s ei(Kc.,,,h-Q.T) _ iKcmh r ei(Kh-ncT,T) _ -iUc,,rwhereT(K,Q) = ￡ bm-------------------:-------, G{K,Q) = ^6n------------------:--------. ItTii—l n=\is a discrete counterpart of (17).Using Theorem 8, by comparing the Preissman box scheme with mostly used schemes such as Euler box scheme and the explicit midpoint scheme, etc., numerical results show that the MSRK is the optimal one.(3) Finally, exploiting the Stormer-Verlet scheme, which is the favorite one in numerical ODE, we provided several practical low order MSRK schemes for Schrodinger equation, semilinear hyperbolic/elliptic equations and nonlinear beam equation. We emphasize that the results in this section can be obviously extended to high-order schemes.References[1] J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114(1994), pp. 185-200.[2] M. Y. Huang, Computational Method for Evolution Equations. Scientific Press, Beijing, 2004. [3J T. J. Bridges, Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil.Soc, 121(1997), pp. 147-190.[4] T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Physic Letter A, 284(2001), pp. 184-193.